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Existence and uniqueness of global solution of nonlinear schrödinger equations onR 2

Existence and uniqueness of global solution of nonlinear schrödinger equations onR 2 No. 4 A ROBUST ALGORITHM FOR OPTIMIZATION PROBLEMS 373 Hence for sufficiently large k E K, from (4.11) we know W~+n0k = 0. On the other hand, from the fact I J+ n Q*I = IQ~I and (4.11) we have I J+ M Q~l -- IQ~l, hence, from the definition of ~, we easily obtain [2k = ~ for sufficiently large k E K. Thus s ~ (k E K) are all defined by (14), which contradicts the fact that s } is all defined by/10) for any k E K. Hence (4.7) is true and the lemma follows. We conclude this paper by proving the following theorem. Theorem 4.3. If (H2), (H3) and (H4) are satisfied, then the infinite point sequence {x k } generated by our algorithm converges superllnearly to the unique optimal solution a:* of (NP). Proof. From (4.11) and Lemma 4.2, Qk is equal to some fixed Q for sufficiently large k. The algorithm in Section 3 is equivalent to the variable metric algorithm described in [8] on the hyperplane {x l aTx ---- bj (j e Q)}. According to the results of [8], we obtain the conclusion of our theorem. References [1] D. Goldfarb. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Existence and uniqueness of global solution of nonlinear schrödinger equations onR 2

Existence and uniqueness of global solution of nonlinear schrödinger equations onR 2

Abstract

No. 4 A ROBUST ALGORITHM FOR OPTIMIZATION PROBLEMS 373 Hence for sufficiently large k E K, from (4.11) we know W~+n0k = 0. On the other hand, from the fact I J+ n Q*I = IQ~I and (4.11) we have I J+ M Q~l -- IQ~l, hence, from the definition of ~, we easily obtain [2k = ~ for sufficiently large k E K. Thus s ~ (k E K) are all defined by (14), which contradicts the fact that s } is all defined by/10) for any k E K. Hence (4.7) is true and the lemma follows. We conclude this paper by proving the...
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Publisher
Springer Journals
Copyright
Copyright © 1998 by Science Press
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02683820
Publisher site
See Article on Publisher Site

Abstract

No. 4 A ROBUST ALGORITHM FOR OPTIMIZATION PROBLEMS 373 Hence for sufficiently large k E K, from (4.11) we know W~+n0k = 0. On the other hand, from the fact I J+ n Q*I = IQ~I and (4.11) we have I J+ M Q~l -- IQ~l, hence, from the definition of ~, we easily obtain [2k = ~ for sufficiently large k E K. Thus s ~ (k E K) are all defined by (14), which contradicts the fact that s } is all defined by/10) for any k E K. Hence (4.7) is true and the lemma follows. We conclude this paper by proving the following theorem. Theorem 4.3. If (H2), (H3) and (H4) are satisfied, then the infinite point sequence {x k } generated by our algorithm converges superllnearly to the unique optimal solution a:* of (NP). Proof. From (4.11) and Lemma 4.2, Qk is equal to some fixed Q for sufficiently large k. The algorithm in Section 3 is equivalent to the variable metric algorithm described in [8] on the hyperplane {x l aTx ---- bj (j e Q)}. According to the results of [8], we obtain the conclusion of our theorem. References [1] D. Goldfarb.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 4, 2007

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